3 research outputs found
Problems and results on 1-cross intersecting set pair systems
The notion of cross intersecting set pair system of size ,
with and
, was introduced by Bollob\'as and it became an
important tool of extremal combinatorics. His classical result states that
if and for each .
Our central problem is to see how this bound changes with the additional
condition for . Such a system is called -cross
intersecting. We show that the maximum size of a -cross intersecting set
pair system is
-- at least for even, ,
-- equal to
if and ,
-- at most ,
-- asymptotically if is a linear hypergraph ( for ),
-- asymptotically if and are both linear
hypergraphs
Dominating sets in Kneser graphs
This thesis investigates dominating sets in Kneser graphs as well as a selection of other topics related to graph domination. Dominating sets in Kneser graphs, especially those of minimum size, often correspond to interesting combinatorial incidence structures.
We begin with background on the dominating set problem and a review of known bounds, focusing on algebraic bounds. We then consider this problem in the Kneser graphs, discussing basic results and previous work. We compute the domination number for a few of the Kneser graphs with the aid of some original results. We also investigate the connections between Kneser graph domination and the theory of combinatorial designs, and introduce a new type of design that generalizes the properties of dominating sets in Kneser graphs. We then consider dominating sets in the vector space analogue of Kneser graphs. We end by highlighting connections between the dominating set problem and other areas of combinatorics. Conjectures and open problems abound
More on maximal intersecting families of finite sets
AbstractNew upper bounds for the size of minimal maximal k-cliques are obtained. We show (i) m(k)⩽k5 for all k; (ii) m(k)⩽34k2 + 32k − 1, if k is a prime power